Symmetries of symmetries and geometrical CP violation

TL;DR

This paper explores transformations that are not symmetries but preserve the set of symmetry elements, revealing physical degeneracies and providing new insights into the origins of calculable phases and geometrical CP violation.

Contribution

It generalizes the concept of symmetry transformations to include equivalence transformations, linking them to physical degeneracies and stationary points in theories with complex symmetry structures.

Findings

Transformations that leave symmetry sets invariant indicate physical degeneracies.

Stationary points form multiplets as representations of equivalence transformation groups.

Stationary points satisfy homogeneous linear equations, aiding potential minimization.

Abstract

We investigate transformations which are not symmetries of a theory but nevertheless leave invariant the set of all symmetry elements and representations. Generalizing from the example of a three Higgs doublet model with $\Delta(27)$ symmetry, we show that the possibility of such transformations signals physical degeneracies in the parameter space of a theory. We show that stationary points only appear in multiplets which are representations of the group of these so-called equivalence transformations. As a consequence, the stationary points are amongst the solutions of a set of homogeneous linear equations. This is relevant to the minimization of potentials in general and sheds new light on the origin of calculable phases and geometrical CP violation.